Creating 2D Pareto Front with Python
Creating 2D Pareto Front with Python
Step by step:
-
Install python 3 via anaconda 3.
- Tip: you will need conda, the package manager that comes with anaconda 3.
-
Install pygmo via conda in anaconda console or in cmd.exe with anaconda directory added to Path environment variable:
- conda config –add channels conda-forge
- conda install pygmo
- Tip: you will need function from pygmo to sort your population using the concept of Pareto dominance.
- Tip: pygmo is a wrapper for C++ library pagmo, which stands for PArallel Global Multi-objective Optimization, developed by people from Europe Space Agency.
-
Before making the plot, one needs something like “swarm data”. To my experience, it is usually an archive of all the individuals evaluated during the optimization. One should prepare this himself. Here, I prepare the swarm data in a format of “free variable as a list + objectives as a list” for one individual.
# swarm_data is a list of lists. See appendix for the full list. swarm_data = [ [free variables, objectives], [free variables, objectives], ...]
- Now we need to sort the object swarm_data using the concept of Pareto dominance.
import pygmo as pg # A dummy class as UDP class Problem_BearinglessSynchronousDesign(object): # Define objectives def fitness(self, x): # evaluate x f1, f2, f3 = 0, 0, 0 return [f1, f2, f3] # Return number of objectives def get_nobj(self): return 3 # Return bounds of decision variables (a.k.a. chromosome) def get_bounds(self): bounds_denorm = [[10.499999999999998, 27.0], [0.5, 5], [35.469240724123004, 53.203861086184496], [112.5, 135.0], [12.942477796076938, 19.413716694115404], [3, 6], [2.5, 7], [27.0, 45.0], [18.0, 21.9375], [9.100179700366597, 13.650269550549895], [38.197186342054884, 57.29577951308233], [2.5, 6], [2.5, 6]] min_b, max_b = np.asarray(bounds_denorm).T return ( min_b.tolist(), max_b.tolist() ) # Return function name def get_name(self): return "Bearingless PMSM Design" # UDP: user defined problem udp = Problem_BearinglessSynchronousDesign() prob = pg.problem(udp) number_of_chromosome = len(swarm_data) pop_archive = pg.population(prob, size=number_of_chromosome) for i in range(number_of_chromosome): pop_archive.set_xf(i, swarm_data[i][:-3], swarm_data[i][-3:])
- Finally, plot using pop_archive.
def my_2p5d_plot_non_dominated_fronts(points, marker='o', comp=[0, 1], up_to_rank_no=1, no_text=True, ax=None, fig=None, no_colorbar=False, z_filter=None, label=None):
# this is adapted from pygmo package but there is a bug therein so I write my own function (I also initiated an issue at their Github page and they acknowledge the issue).
full_comp = [0, 1, 2]
full_comp.remove(comp[0])
full_comp.remove(comp[1])
z_comp = full_comp[0]
fronts, _, _, _= pg.fast_non_dominated_sorting(points)
# We define the colors of the fronts (grayscale from black to white)
cl = list(zip(np.linspace(0.9, 0.1, len(fronts)),
np.linspace(0.9, 0.1, len(fronts)),
np.linspace(0.9, 0.1, len(fronts))))
if ax is None:
fig, ax = plt.subplots(constrained_layout=False)
plt.subplots_adjust(left=None, bottom=None, right=0.85, top=None, wspace=None, hspace=None)
count = 0
for ndr, front in enumerate(fronts):
count += 1
# Frist compute the points coordinates
x_scale = 1
y_scale = 1
z_scale = 1
if comp[0] == 1: # efficency
x_scale = 100
if comp[1] == 1: # efficency
y_scale = 100
if z_comp == 1: # efficency
z_scale = 100
x = [points[idx][comp[0]]*x_scale for idx in front]
y = [points[idx][comp[1]]*y_scale for idx in front]
z = [points[idx][z_comp] *z_scale for idx in front]
# # We plot the points
# for idx in front:
# ax.plot(points[idx][comp[0]], points[idx][comp[1]], marker=marker, color=cl[ndr])
# Then sort them by the first objective
tmp = [(a, b, c) for a, b, c in zip(x, y, z)]
tmp = sorted(tmp, key=lambda k: k[0])
# Now plot using step
ax.step([coords[0] for coords in tmp],
[coords[1] for coords in tmp], color=cl[ndr], where='post')
# Now add color according to the value of the z-axis variable usign scatter
if z_filter is not None:
z = np.array(z)
x = np.array(x)[z<z_filter]
y = np.array(y)[z<z_filter]
z = z[z<z_filter]
print('Cost, -Efficency, Ripple Sum')
for a,b,c in zip(x,y,z):
print(a,b,c)
scatter_handle = ax.scatter(x, y, c=z, edgecolor=None, alpha=0.5, cmap='Spectral', marker=marker, zorder=99, vmin=0, vmax=20, label=label) #'viridis' Spectral
else:
scatter_handle = ax.scatter(x, y, c=z, edgecolor=None, alpha=0.5, cmap='Spectral', marker=marker, zorder=99) #'viridis'
if not no_colorbar:
# color bar
cbar_ax = fig.add_axes([0.875, 0.15, 0.02, 0.7])
cbar_ax.get_yaxis().labelpad = 10
clb = fig.colorbar(scatter_handle, cax=cbar_ax)
else:
pass # use vmin/vmax or set_clim
# if comp[0] == 0 and comp[1] == 1:
# min_, max_ = 0, 20 # Only implemented for Ripple Sum
# scatter_handle.set_clim(min_, max_) # https://stackoverflow.com/questions/3373256/set-colorbar-range-in-matplotlib
# # # # color bar
# # # cbar_ax = fig.add_axes([0.875, 0.15, 0.02, 0.7])
# # # cbar_ax.get_yaxis().labelpad = 10
# # # clb = fig.colorbar(scatter_handle, cax=cbar_ax)
# # # clb.ax.set_ylabel('Ripple Sum [1]', rotation=270)
# else:
# raise Exception('Not Implemented')
if z_comp == 0:
z_label = r'$\rm {Cost}$ [USD]'
z_text = '%.0f'
elif z_comp == 1:
z_label = r'$-\eta$ [%]'
z_text = '%.1f'
elif z_comp == 2:
z_label = r'$O_C$ [1]'
z_text = '%.1f'
if not no_colorbar:
clb.ax.set_ylabel(z_label, rotation=270)
if z_comp == 2: # when OC as z-axis
print('-----------------------------------------------------')
print('-----------------------------------------------------')
print('-----------------------------------------------------')
# Add index next to the points
for x_coord, y_coord, z_coord, idx in zip(x, y, z, front):
if no_text:
pass
else:
ax.annotate( z_text%(z_coord) + ' #%d'%(idx), (x_coord, y_coord) )
else:
# text next scatter showing the value of the 3rd objective
for i, val in enumerate(z):
if no_text:
pass
else:
ax.annotate( z_text%(val), (x[i], y[i]) )
# refine the plotting
if comp[0] == 0:
ax.set_xlabel(r'$\rm {Cost}$ [USD]')
elif comp[0] == 1:
ax.set_xlabel(r'$-\eta$ [%]')
elif comp[0] == 2:
ax.set_xlabel(r'$O_C$ [1]')
if comp[1] == 0:
ax.set_ylabel(r'$\rm {Cost}$ [USD]')
elif comp[1] == 1:
ax.set_ylabel(r'$-\eta$ [%]')
elif comp[1] == 2:
ax.set_ylabel(r'$O_C$ [1]')
ax.grid()
# plot up to which domination rank?
if up_to_rank_no is None:
pass
else:
if count >= up_to_rank_no:
break
# fig.savefig(r'C:\Users\horyc\Desktop/'+ '2p5D-%d%d.png'%(comp[0],comp[1]), dpi=300)
return scatter_handle
def my_plot(fits, vectors):
my_2p5d_plot_non_dominated_fronts(fits, comp=[0,1], marker='o', up_to_rank_no=1)
my_2p5d_plot_non_dominated_fronts(fits, comp=[0,2], marker='o', up_to_rank_no=1)
my_2p5d_plot_non_dominated_fronts(fits, comp=[1,2], marker='o', up_to_rank_no=1)
my_plot(pop_archive.get_f(), pop_archive.get_x())
Appendix
The complete swarm data:
swarm_data = [
[47.6279, 2.56821, 23.1075, 134.89762000000002, 33.5248, 3.6783, 5.06812, 119.748, 17.5046, 55.99681999999999, 2.53512, 181.519, -0.894019, 32.9708],
[35.6079, 1.37345, 27.505, 125.92096999999998, 33.3697, 3.33328, 3.44401, 113.72, 18.4888, 50.00361, 3.44401, 176.281, -0.821388, 60.8382],
[34.3172, 0.811953, 20.9468, 112.29583, 33.4988, 3.60395, 2.7315, 146.71, 16.323, 47.2029, 2.7315, 183.779, -0.800641, 43.3236],
[53.5207, 1.33942, 27.2844, 128.33282, 32.1828, 4.43271, 3.78626, 122.483, 16.4472, 50.02606, 3.78626, 176.727, -0.913629, 43.455],
[40.1099, 2.65872, 21.1027, 134.91009, 29.0121, 5.58964, 5.60804, 161.562, 21.5467, 48.93456, 4.32716, 194.71, -0.898801, 32.2833],
[35.8012, 3.67623, 19.681, 126.02125, 28.0372, 3.61269, 6.84931, 154.107, 19.1896, 39.21851, 3.65771, 200.113, -0.867071, 36.4768],
[51.121, 4.57399, 21.5483, 123.57468, 37.5238, 4.33728, 2.9666, 122.691, 21.8315, 47.184799999999996, 2.9666, 156.543, -0.928768, 35.7334],
[44.7841, 1.61475, 20.3023, 123.25583, 28.0818, 5.3399, 3.34562, 154.11, 18.2395, 57.23464, 2.93494, 189.295, -0.844009, 37.9452],
[41.5561, 2.11055, 20.2725, 119.18502000000001, 26.9402, 4.83056, 2.8033, 155.683, 15.0338, 53.3397, 2.7004, 183.159, -0.841678, 36.4282],
[49.7316, 4.99837, 24.1713, 127.17805, 35.3797, 4.50703, 2.92772, 131.355, 17.5577, 47.11911, 2.91801, 161.707, -0.918028, 33.2947],
[43.0948, 3.51911, 22.133, 125.17177, 37.3387, 4.09355, 4.22781, 123.414, 20.3289, 50.38821, 2.90831, 166.102, -0.893833, 36.0308],
[28.2913, 4.63508, 19.4072, 131.78133, 36.3657, 3.68546, 6.29176, 129.022, 20.2252, 41.6477, 4.1815, 226.188, -0.809318, 35.5877],
[49.4155, 3.76259, 27.4126, 131.25297999999998, 34.0473, 4.55706, 5.36665, 125.116, 17.4152, 53.15101, 2.84091, 191.043, -0.917317, 23.4887],
[40.0233, 2.8821, 20.3747, 119.04965, 33.3239, 3.90521, 3.99764, 121.741, 18.3559, 55.932939999999995, 3.99764, 175.735, -0.836569, 50.652],
[23.8001, 1.49918, 19.1734, 120.99667000000001, 29.8978, 3.05498, 4.59157, 135.631, 17.771, 49.65559, 3.30849, 222.875, -0.658562, 48.1593],
[43.1509, 3.45284, 26.2934, 130.09985999999998, 29.1796, 5.26936, 4.27969, 161.661, 18.6098, 42.18459, 4.27969, 174.646, -0.933547, 30.2867],
[51.6553, 2.20242, 20.3405, 134.50934, 30.717, 3.32957, 6.69236, 166.337, 18.6737, 40.905229999999996, 4.15403, 198.067, -0.953787, 34.753],
[28.3507, 3.34434, 22.2174, 125.2636, 35.2307, 4.85973, 6.89977, 149.696, 17.7309, 55.80508, 4.87638, 238.157, -0.749254, 42.6059],
[28.1042, 3.4309, 25.8944, 138.10625, 26.9681, 5.40681, 4.33637, 146.897, 19.3461, 49.55997000000001, 4.33637, 202.522, -0.831739, 38.4119],
[25.185, 3.51215, 20.3372, 126.17133, 32.6982, 5.83568, 4.26376, 127.003, 18.4787, 38.5636, 3.7342, 220.699, -0.917799, 29.4485],
[26.4724, 3.8235, 24.5342, 121.43704, 36.3612, 3.22706, 5.13046, 113.881, 20.7696, 44.41676, 5.13046, 184.857, -0.811968, 66.0435],
[30.6682, 2.79631, 21.0116, 129.16507000000001, 32.2481, 5.09019, 5.06881, 159.714, 15.2094, 49.58271, 5.06881, 225.953, -0.813221, 39.5435],
[45.2299, 1.96466, 18.5294, 131.43649, 33.1744, 3.13818, 5.06135, 129.539, 17.7051, 53.48375, 5.06135, 185.602, -0.868739, 42.5802],
[26.6444, 3.11113, 22.3387, 119.11159, 26.8862, 5.80115, 2.91025, 143.959, 19.5829, 56.57994999999999, 2.91025, 206.947, -0.770347, 41.6247],
[43.4229, 1.04924, 22.3563, 130.05476, 33.4765, 3.91315, 4.05282, 175.224, 18.8214, 40.97991999999999, 4.05282, 195.512, -0.913315, 36.9309],
[31.6615, 2.28778, 24.0682, 126.17217, 32.4538, 5.57476, 3.52022, 164.902, 15.6513, 39.34108, 3.28028, 202.406, -0.908245, 34.7973],
[47.7807, 2.59304, 24.1481, 127.69185999999999, 26.6133, 4.48501, 6.14776, 153.595, 20.8398, 45.980579999999996, 3.92238, 175.238, -0.93631, 29.0386],
[23.9522, 4.51404, 20.0511, 134.62125, 27.3415, 5.62565, 3.04345, 158.406, 20.7828, 57.23675, 3.04345, 234.122, -0.692232, 47.2431],
[47.7607, 2.66568, 23.9515, 129.82992000000002, 26.3372, 4.1764, 2.63563, 120.653, 18.1829, 52.45903, 2.63563, 172.807, -0.868553, 42.7441],
[49.5071, 4.87094, 22.327, 128.36201, 35.1236, 5.03911, 4.25722, 130.392, 20.6247, 53.245, 2.8154, 170.206, -0.920707, 26.312],
[40.766, 3.49232, 19.8596, 126.77597, 35.2342, 3.54629, 5.63557, 171.536, 18.8965, 48.00559, 4.40109, 211.561, -0.845613, 45.7064],
[38.0565, 3.72951, 19.8923, 124.50897, 38.7463, 3.10008, 3.77744, 114.634, 19.2404, 43.10804, 3.77744, 188.114, -0.85624, 54.0804],
[35.5119, 0.701105, 24.7364, 123.96596, 33.2149, 3.69437, 6.44474, 149.923, 21.5765, 56.23407, 3.87477, 219.84, -0.717125, 46.1003],
[39.0981, 4.60562, 23.8122, 123.30583, 35.6233, 4.04023, 4.545, 159.474, 21.5746, 51.94286, 3.94586, 183.701, -0.824435, 35.4473],
[47.2003, 2.09601, 21.8559, 127.54092, 38.6635, 5.44767, 2.94258, 114.387, 14.7743, 44.379979999999996, 2.94258, 190.292, -0.940096, 32.7037],
[45.8647, 0.844416, 25.0245, 131.90462000000002, 28.458, 5.76641, 4.71662, 165.774, 17.3464, 41.96275, 2.54755, 190.13, -0.926318, 35.1369],
[21.6498, 3.26922, 26.2285, 133.92583000000002, 35.9298, 4.62049, 5.94586, 119.286, 15.6634, 55.358410000000006, 3.54231, 223.567, -0.783692, 43.8435],
[49.3674, 0.931327, 21.4399, 121.23339000000001, 30.1177, 5.77861, 5.10071, 167.043, 16.0343, 51.938790000000004, 3.84299, 211.409, -0.880342, 39.3778],
[47.1324, 4.27181, 19.7131, 127.48852, 33.7363, 5.83162, 4.56808, 116.641, 15.8566, 39.48504, 4.51154, 194.7, -0.962088, 30.2462],
[37.412, 1.31527, 25.5934, 124.8741, 26.7538, 4.75757, 2.71002, 115.041, 20.362, 50.487719999999996, 2.71002, 180.538, -0.848597, 51.9382],
[41.657, 3.41631, 23.3138, 131.68966, 37.5834, 5.21593, 4.08877, 119.149, 15.812, 44.95617, 4.08877, 183.303, -0.929572, 36.4526],
[46.5669, 2.31713, 18.8458, 125.49558999999999, 26.3901, 4.69822, 2.80369, 156.548, 19.1164, 41.67169, 2.80369, 175.378, -0.935941, 28.922],
[24.6114, 4.71183, 25.0337, 118.82785000000001, 26.1804, 4.36504, 3.52257, 170.458, 19.5376, 47.97367, 3.52257, 205.801, -0.710573, 51.9039],
[34.4651, 1.58984, 22.8437, 126.12086, 37.7644, 3.35433, 4.36026, 113.274, 17.5438, 51.71744, 3.82394, 187.467, -0.793467, 55.1992],
[30.6327, 2.44905, 22.2462, 125.09318000000002, 30.814, 4.04997, 2.9522, 141.569, 16.8527, 42.1351, 2.9522, 186.36, -0.857521, 38.1999],
[35.9367, 4.79408, 21.0845, 139.83882, 34.5988, 5.28571, 4.20396, 132.151, 17.1602, 54.92416000000001, 4.20396, 198.525, -0.851245, 42.6942],
[45.0811, 0.595161, 22.0219, 120.99614100000001, 37.8653, 3.77345, 5.08561, 134.719, 18.6577, 50.59486, 3.02696, 185.206, -0.853774, 32.782],
[37.8768, 4.89898, 22.7729, 123.36655999999999, 27.9718, 3.0759, 4.45014, 161.317, 15.736, 50.08517, 2.95157, 191.933, -0.788985, 38.4905],
[24.4619, 3.82924, 22.2931, 136.08366, 34.565, 3.86195, 5.27352, 131.635, 20.8572, 50.60974, 4.85844, 212.186, -0.744519, 52.8565],
[40.5667, 4.35503, 20.1728, 129.41885, 31.5068, 4.9683, 5.37349, 175.443, 17.0767, 50.789590000000004, 5.37349, 210.455, -0.881344, 40.7006],
[48.9763, 1.16851, 19.3764, 113.85346, 32.7069, 5.6488, 2.91293, 156.365, 16.7672, 53.38453, 2.91293, 183.075, -0.890456, 33.0605],
[50.074, 2.07716, 25.7412, 132.27774, 30.3896, 5.77965, 3.61842, 145.288, 21.7922, 44.51489, 2.88559, 172.726, -0.942986, 25.6144],
[37.291, 3.10691, 20.486, 131.66537, 34.9566, 3.07602, 3.03269, 148.568, 21.5796, 44.32369, 3.03269, 185.926, -0.816715, 45.5186],
[24.1716, 4.0044, 27.1528, 126.10170000000001, 35.1673, 3.93588, 2.54781, 127.412, 19.5975, 56.35481, 2.54781, 210.311, -0.704616, 64.1407],
[49.2344, 1.69912, 27.4563, 127.06137999999999, 31.8166, 3.34645, 2.7531, 159.443, 15.9932, 45.4153, 2.7531, 164.19, -0.882498, 34.5574],
[52.3444, 2.09733, 22.0843, 130.07420000000002, 26.9175, 3.852, 3.65896, 167.702, 21.6681, 39.04562, 3.37922, 162.643, -0.952239, 28.5796],
[41.6171, 0.927392, 19.0635, 115.92438999999999, 37.1571, 3.22161, 2.61604, 176.675, 16.6682, 53.19374, 2.61604, 202.761, -0.736894, 50.1187],
[40.6204, 1.81361, 22.248, 131.51292, 34.85, 3.47069, 5.01165, 147.205, 16.9675, 50.36049, 3.56389, 192.623, -0.844965, 35.3137],
[32.1196, 4.04912, 23.6792, 120.99858, 33.8394, 4.36704, 4.98256, 158.42, 16.0969, 40.434020000000004, 3.89582, 192.994, -0.873164, 36.1708],
[40.8722, 3.59484, 19.1644, 123.75306, 36.2405, 3.66307, 3.39868, 138.735, 17.9146, 47.28358, 3.39868, 177.853, -0.865922, 38.9565],
[22.1289, 2.89755, 25.2635, 126.19603000000001, 33.5194, 3.06945, 3.33727, 173.002, 16.1631, 48.23527, 3.33727, 224.402, -0.634841, 57.1599],
[22.7962, 2.40996, 23.6704, 123.93874, 28.2301, 3.27922, 3.40364, 148.899, 16.9091, 46.95044, 3.40364, 199.713, -0.708937, 46.4991],
[28.4809, 2.58296, 26.8277, 120.64044, 38.1392, 3.85753, 6.1323, 139.375, 17.5568, 47.30139, 3.57379, 197.67, -0.821003, 33.8732],
[26.5901, 3.86492, 22.3524, 127.14359, 37.3533, 3.1385, 3.30084, 165.205, 20.7437, 52.800360000000005, 2.93076, 240.716, -0.547709, 63.5368],
[39.1509, 2.52324, 20.7846, 134.46256, 35.9106, 3.44958, 5.26156, 119.601, 19.0514, 40.136280000000006, 4.97558, 202.673, -0.907558, 43.702],
[25.5959, 1.4491, 25.2004, 125.93145, 26.2015, 5.79975, 2.7265, 146.263, 18.1558, 40.6665, 2.7265, 195.142, -0.903368, 28.9441],
[32.1347, 2.0326, 24.7534, 115.1842, 38.2808, 4.66567, 4.7175, 155.303, 17.899, 42.066939999999995, 4.14494, 193.45, -0.891558, 34.6746],
[39.8273, 2.37637, 26.8892, 122.10556, 30.0341, 5.71614, 4.37003, 125.439, 17.9333, 39.49903, 4.37003, 171.947, -0.949681, 33.3721],
[38.2415, 4.45267, 21.3365, 121.6559, 32.4562, 3.73516, 3.93592, 162.788, 20.7408, 43.705189999999995, 3.75919, 176.007, -0.85311, 36.6412],
[38.6151, 1.67432, 18.9285, 121.68768, 34.1082, 5.89017, 3.35483, 119.033, 20.2911, 55.930670000000006, 2.87557, 187.483, -0.865058, 40.5336],
[23.7534, 4.80729, 18.5203, 130.84454, 35.7392, 4.39946, 5.35984, 123.307, 20.9343, 54.21794, 5.35984, 233.316, -0.706834, 62.4222],
[40.5379, 2.49642, 24.5291, 127.36363, 32.0251, 3.74693, 2.68977, 154.004, 14.8984, 41.36197000000001, 2.68977, 170.431, -0.891656, 36.613],
[42.4656, 3.61241, 21.7163, 129.77941, 36.7871, 5.90617, 6.65398, 172.339, 15.7214, 48.32691, 4.00731, 230.629, -0.920714, 39.5745],
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]
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